Can feature selection be considered a way to observe relationship between variables like correlation? In correlation we can observe relationship between a pair of variables, let me call it X1 and Y. 
Now, considering I have the predicting variables X1, X2, ..., Xn and the variable Y. Does the following assumption holds: 
If the variables Xi,...,Xj are observed to better classify Y, where 1 <= i,j <= n, can I claim that the variables Xi,...,Xj have more association with Y rather the remaining variables Xk, k not belonging to the i,j interval. 
 A: In a word yes. In a Bayesian framework, a good feature  (or feature set) $X$ is one where $P(Y | X )$ depends significantly on the specific value of $X$ observed; which can be reasonably interpreted as the variables being correlated. One way to quantify this is in terms of the relative entropy between the distributions.
A: A variable that achieves a better classification performance is not necessarily a variable that is correlated or relevant to the target class labels (i.e. Y in your case). So, optimality does not imply relevance and vice versa.
If the relevance metric you are using does not apply any transformation or pre-processing to the data and independent of any learning model, then you can infer some direct relation between the selected feature and the target variable. A good metric is one that leads to selection of features related to the target variable and that are not redundant. mRMR is an example of such a metric.
On the other hand, if your evaluation metric is not independent of the classification model which is known as the wrapper model, then selection of relevant features is based on the type of the classifier. For example, Naive Bayes Classifier and Decision Trees are methods that reflects relevance between the feature and the target variable. However, SVM with an RBF kernel might not lead to the selection of relevant features but rather, the optimal ones.
